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Possible to Self Teach Adanced Mathematics

  1. Jul 30, 2008 #1
    Let me explain. I have bachelors in pure math (as opposed to applied math) and a bachelors and masters in nuclear engineering. I am employed as a nuclear engineer and I am happy. Before completing my masters, I did one semester of graduate work in math (I might still be there had I not taken combinatorics, differential topology, linear algebra, and complex analysis all in the same semester without the best background in math).

    But anyway, I would like to continue to learn math (in particular I would really want to learn more number theory), but I cannot go back to school.

    I was wondering what some of you think of my predicament, and whether you guys could supply either some ideas of how to complete some further study in graduate mathematics without, seemingly, the necessary resources. I would also be interested if anyone knows about some open course ware similar the the MIT program.

  2. jcsd
  3. Jul 31, 2008 #2
    Are they not analysis I orII WHICH i supposed you did in your Bachelors advanced mathematics and certainly of the highest if not the highest level among all other subjects???
    You mean you jumped over a 3 meters obstacle and now you ask advise how you going to step outside your house?????
  4. Jul 31, 2008 #3
    Not exactly. You are right, I have taken a few classes on analysis and some of the other advanced subjects in math, but consider I was studying how to drink water. Yes, I may have drunk a few glasses of water, maybe even a pitcher, but I want to know how to drink Lake Superior. Maybe not the best analogy, but I hope I get my point across.
  5. Jul 31, 2008 #4
    If you want to drink Lake Superior you can do it the following way:
    Double the amount of water glasses that you drank the previous day,staring with two glasses the 1st day.
    Can you calculate how much water(expressed in lakes or rivers) you must have drunk in lets say 12 months????????
  6. Dec 21, 2008 #5
    Of course, it is possible to Teach Yourself Advanced anything. The problem is our society has turned education to be the means to an end, instead of a continuation of an unending journey.
    If you ask me most degrees expire within a year or two, once people get that Masters or PhD, find a cushy job that's it. Either they become specialists in one thing or they focus on promotions and more pay, not learning.
    I am not a mathematician, but I do enjoy maths as you know maths is basically the glue that holds all other disciplines together within the different taxonomies, as well math is the path that uncovers the relationships (or differences) of the many areas of studies or of arts and sciences, okay, and mathematics. Therefore the vastness of the subject can be mind boggling, however the depth within a containable area is also of the quantum level, therefore you could happily drown in mathematics for the rest of your life. You do not need anyone to teach you because at some point there is no one who can teach you, perhps discuss some concept. For example for the past 30 years, I only met one young many about 2o years ago who happened to sit next to me in a flight from Houston to Chicago. As was customary thoe days between a white and a black guy we shared deep silence. Until he noticed my pharmacist type notebook that I was scribbling in with cryptic formulae, to my surprise he was able to relate to what I was doing, I was trying find a solution that satisfies x3=b3+c3!
    and he said whay are you even doing that when half the PhD's in the world would get 2+2X2=X wrong! He was quite smart, he later told me his occupation. He was a Mathematic PhD. research scholar at the university of Chicago.
    I woul late test his 2=2X2 =x theory with a bunch of Chicago High School Teachers that joined my Computer Literacy Class I taught for the Chicago City Wide College, granted not all of them were PhD's, but they were all college graduates more than 1/2 the class got the problem wrong, and tried to justfy their wrong answers in extremely unmathematical ways.
    So yes, you can teach yourself, and you will be pleasantly surprised at what you discover. For example, I can categorically say that earth will perish, I do not even know if perish is the right word, but perhaps will die one day, but contrary to popular belief it is not going to be in 2012. Lets just say mathematically we will know the end of the plannet thousands if not millions of years before it happens, why? the universe is the king of normal distribution. Planet earth is not within the Q range of its life-cycle. And sorry Al Gore mathematics show that global warming has very little to do with what humans are doing. Maybe pollution yes, but global warming? Maths says we are simply not that significantly important to modulate.
    So live your life, earth is here for a while, not I am not saying humanity - because a few events might wipe us out including our own self destruction, but earth will be here quite healthy for that matter. So pick up a book on princepia mathmatica.
    There many Interesting topics from many university. Bu my favorite one since zi moved from Chicago to Jersey (Dumb Move). is a mixture of pure and applied maths readings from:
    "[URL [Broken]
    Good luck I also have a suspicion that the largest number and the smallest number are one and the same differentiated only by direction, if I could prove that, we could open the door to a new beginning.
    Last edited by a moderator: May 3, 2017
  7. Dec 22, 2008 #6


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    just take one good book at a time and read it and work the exercises, and think abiutn it.

    just to recap and correct your waterloo experience,

    start with perhaps hoffman and kunze in linear algebra.

    then perhaps cartan in complex analysis.

    then guillemin and pollack in differential topology if apropriate, or better, milnor.
  8. Dec 22, 2008 #7


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    i myself, when out teaching math in a college and wanting to go back to grad school taught carefully through spivak's calculus, and his calculus on manifolds, then read cartan's complex variables, and worked a lot of problems in big rudin and herstein's topics in algebra, but probably hoffman and kunze would have been better for linear algebra.

    dummitt and foote is also a good choice for learning algebra today, in spite of my local criticisms of parts of it. especially if you work the problems faithfully.
  9. Dec 23, 2008 #8
    Find a local (probably best if it's private) university around the middle of August or the very beginning of January, and find their bookstore. You may just be able to buy some advanced math books. It will be pricey, but it will provide you with the necessary textbooks.

    Also, it is probably helpful if you do some internet research, first, to check for the necessary prerequisites to the courses you are interested in, and buy those books, too.

    I do a fair amount of self-teaching. I've taught myself the larger aspects of tensor theory, and it took about a year of my spare time searching through books (and my abstract linear algebra course didn't hurt, either) that until it made sense.

    But yes, it is possible to teach yourself high end things.
  10. Jan 7, 2009 #9
    There's quite an easy way to actually approach this. At my undergraduate institution, I thought our first year courses in analysis were too simple so in the spring I found through google the following page:


    (by changing the "a" after 55 to "b" you get the follow-up course.) This is Harvard's honors calculus sequence and it has some pretty good problems. Another Harvard prof. has a great page too:


    You can look at Harvard's course catalog to figure out the prerequisites for the different courses. This way you can easily build a plan for you to study on your own. I worked through 4 of Curt McMullen's courses during my undergraduate by reading the books in his syllabus and doing all the assigned exercises. For some of the courses you even have the solutions posted, so it helps a lot if you get stuck for hours on a problem.

    If you go to Caltech's website you can also find the list of their first year graduate courses. Just look at the course code e.g. 120a for algebra and google "algebra 120a site:caltech.edu". You'll most probably find the course page with all the homework and solutions and books. Then you can pretty much work through their program. Of course, you'll miss all the discussions in class about how the material relates to current research etc. but it's a pretty good way to study on your own. Depending on your mathematical capacity, you'll probably hit a brick wall at some point. For example building a good intuition in algebraic geometry is one of the things where it doesn't hurt to have someone to talk to, but at least you can choose your pace, so there's no need to worry.
  11. Aug 8, 2011 #10
    After working through Sergei Treil's wonderful book: "Linear Algebra Done Wrong", I would like to study Advanced Algebra. It seems that the most commonly used texts on the undergraduate level are Dummit and Foote (used as a follow up to Apostol's Calculus vol. 1 and vol. 2 at Caltech), Artin's "Algebra" (used by Harvard and MIT) and Herstein's "Topics in Algebra."

    Which would you recommend for self-study?
  12. Aug 8, 2011 #11
    I agree with the analogy. If you did well in Analysis (from a respectable school.) It's questionable why you would even post this topic in regards to number theory.
    I taught myself number theory over a summer. I just bought a book and had fun. I also brought some questions to a professor that was eager to help when I got stuck.
    Intermediate Analysis is not comparable.
  13. Aug 11, 2011 #12
    Hi Curtis. I joined to respond to you. Artin's Algebra is a good text for self-teaching and you will have the benefit of Benedict Gross's lectures at the Harvard Extension school, free online, which follow that text; you will also have the benefit of MIT OCW materials from Artin's own classes. I like to have a little structure, recommendations on which exercises to do, aural recapitulation of concepts I'm reading, lecture notes that explain the book in a slightly different manner, etc., and so it's very useful for me to have these resources.
  14. Aug 11, 2011 #13
    Setec: Thank you very very much.

    The class looks great. Although, I am a bit apprehensive as the prerequisite for the class at Harvard is Math 25 or 55 which cover Rudin, Spivak-Manifolds, and Axler LADR.

    I studied "Linear Algebra Done Wrong," a more rigorous version of Axler's book used at Caltech, so I am not worried about the Linear Algebra.

    However, at MIT, Professor Artin lists Introductory Analysis (once again Rudin) as a prerequisite. So, while I am studying a nice introductory Analysis book called "Elementary Analysis, the Theory of Calculus" by Ross; this is clearly not Rudin.

    I looked at the first few pages of each book on Amazon, and I really like Herstein's style.

    So, I guess my options are either to get Herstein's Topics in Alegbra, or wait until I study Rudin before taking the online class.
  15. Aug 11, 2011 #14
    Hey again, Curtis. To be honest, I see no reason why analysis would be required for this course. I am working on Rudin concurrently myself, but I don't think it's really been at all necessary. The topics feel quite different at this introductory level. Some familiarity with linear algebra will be helpful, but Artin actually provides an abbreviated introduction to that as well.

    I believe I looked at Herstein briefly, but I'm afraid I don't remember anything about it. It has a very good reputation, though, and I doubt you'll be dissatisfied with either book!
  16. Aug 11, 2011 #15

    Thanks for information. What Math did you study prior to Rudin and Artin?
  17. Aug 11, 2011 #16
    Honestly, very little since high school calculus close to a decade ago. I worked through some of MIT's Multivariable Calculus and Linear Algebra courses to get myself back into the habit of doing math, and I studied a tiny bit of differential equations. Honestly, both books are difficult, but I don't think they're difficult in the way of requiring a lot of background, at least so far. They've simply required more thought than other math I've done (really, than almost anything else I've done!).
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